Basie was a popular figure in the UK and a measure of his band's brilliance was that when they first performed in London, in 1954, they arrived to find that all their printed sheet music had been sent in error to South America. Nobody was able to tell, because the 13-piece band soaring majestically on favourites such as Jumpin' at the Woodside, Li'l Darlin' and April in Paris. It fitted Basie's philosophy of music: "I wanted 13 men to think and play the same way. I wanted the trumpets and trombones to bite with real guts, to be tasty and subtle."

Basie, an only child who was christened William, started out to be a drummer but switched to piano before he was even a teenager. It was in Harlem that he found his first great influence: Fats Waller. He used to literally sit at Waller's feet, at the old Lincoln Theatre in Harlem, watching him work the pedals on the organ and using his hands to imitate the great jazz man. "Then I sat beside him and he taught me the keys," Basie used to recall fondly.

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Through Waller, Basie found a job as an accompanist with a vaudeville act called Katie Crippen and Her Kids. He later joined a touring show headed by Gonzel White, playing piano in a four-piece band.

One event from the Twenties that marked him for life was being taken to hospital in September 1927 with viral meningitis. He was there for a month and felt abandoned by his fellow band members. When he came out he had to take a job as an cinema organist for silent movies but it left him with a lifelong shortage of sympathy for band members with health problems.

It was in the Thirties, when he took over Bennie Moten's elegant boogie-influenced band following the bandleader's sudden death, that he found his creative breakthrough with a band that included young tenor saxophone maestro Lester Young. It was in this band that a swing sound that would change jazz was born. Basie often practised with just his innovative rhythm section, an approach that drove a swing drive that wowed audiences and made him one of the pre-eminent bandleaders of the Big Band era in the Thirties and Forties. President Ronald Reagan once described Basie as "among the handful of musicians that helped change the path of American music".

During the Forties, many of the great jazz postwar musicians passed through the band, including Illinois Jacquet, Don Byas, Wardell Gray, Lucky Thompson and Clark Terry, and the Basie band played at President John F Kennedy's inaugural ball. Jazz historian and trumpeter John Chilton said: "Count Basie had an incredibly economical and delicate piano style that enabled him to be the perfect bandleader. He was a true pioneer but something of a taciturn man. I met him a few times in London and New York and he was always friendly but never overly chatty. There was one famous press conference where an interviewer asked a ponderous and complicated question that took several minutes to deliver, about whether there were African influences in his music, and Basie just replied: 'I guess so'."

A simple approach reflected Basie's ethos. "It's the way you play that makes it. Play like you play. Play like you think, and then you got it."

And how did he get his name? In 1936, when he was playing at the Reno Club in Kansas, the announcer joked that with Earl Hines and Duke Ellington doing so well, Basie needed a better stage name. "Well that was the last time I was ever introduced as Bill Basie," the bandleader said.

Basie was the first African-American male recipient of a Grammy Award and his music continues to inspire young jazz players. And when your records have been chosen for Desert Island Discs by fellow musicians of the calibre of Tony Bennett, Nelson Riddle and Buddy Rich, you can be sure you created a legacy that will survive.